Volumen II - SAM

Congreso SAM/CONAMET 2009 Buenos Aires, 19 al 23 de Octubre de 2009
In order to validate the procedure, a number of spectra from molybdenum at high temperatures were
analyzed. Molybdenum, a group VI of transition metal has a melting point of 2883 K, a high specific heat,
and good corrosion and creep resistance. The melting point of molybdenum is exceeded only by tungsten and
tantalum, among the useful high temperature metals. This metal is ductile at room temperature, with a brittleductile
transition temperature significantly lower than that of tungsten. Molybdenum has also good strength
at high temperatures, being lighter than tungsten and tantalum [1, 2]. These qualities make molybdenum
attractive for the use in the nuclear industry [3 - 6].
Given the generality of the assumptions, the method can be also suitable to other kinds of materials and
relaxation processes, for example, dielectric permittivity or magnetic relaxation.
2. THEORY
The most general expression for a mechanical dynamic relaxation process, such as the dynamical mechanical
modulus M*, when there are two different mechanism involved can be written as:
M P
B
* ( ω , T ) = M * ( ω,
T ) + M * ( ω,
T )
(1)
where M*P is the peak contribution and M*B represents the background. Usually, the term M*P is associated
to a specific imperfection in the material matrix, i. e., the presence of point defects, dislocations, etc.. In this
way, it would be possible, in an ideal case, to neglect this contribution. In such circumstances the
background would be the only term which contribute to the modulus.
However, given the contribution of different mechanisms that affect the matrix, (like vacancies movements),
the background is critically modified in a wide range of frequencies or temperatures, and these modifications
are represented by M*P. In most cases, it is desirable to isolate the contribution from these imperfections to
analyze them separately, even though they do not exist independently of the material matrix.
The natural method to obtain the function M*P from the measured modulus M* is to subtract the background
M*B. Unfortunately, it is not always possible to measure the background, since this requires a sample free of
additional relaxation processes that the ones related to the damping background. Therefore, it is necessary to
develop a procedure to estimate the background in the measured range of interest, with the minimum
assumptions in its functional form.
As M* is a complex number, the analysis can be done in the real part, the storage modulus M1, or in its
imaginary part, the loss modulus M2, or in both. However, M1 is affected by asymptotic values for very low
and very high temperatures (or very high and very low frequencies), which usually must be measured in a
different kind of experiment. On the other hand, the loss modulus is independent of these parameters, and in
consequence is more adequate to perform the background estimations and subtraction.
From eq. (1),
M 2 ( 2P
2B
ω , T ) = M ( ω,
T ) + M ( ω,
T )
(2)
where M2P is the imaginary part of the peak contribution and M2B is the imaginary part of the background. In
most cases, specially when modulus is measured in function of temperature, the function M2B is
monotonically increasing while M2P is a peak that goes to zero far from its maximum. Figure 1 shows a
typical behavior. The background seems to be the side of a peak, since it could be associated to a distribution
of relaxation processes, for example dislocation movement. Therefore, M2B can be represented by a Cole-
Cole function [7], with a relaxation time which depends on temperature as an Arrhenius law,
δM
B
M 2B ( ω,
T ) = Im[
]
(3)
α
⎛ ⎛ H B ⎞⎞
1+
⎜iωτ
0B
exp⎜
⎟⎟
⎝ ⎝ kT ⎠⎠
where δMB is the relaxation magnitude and HB is the energy associated to the background. The parameter α
represents, in the Cole-Cole function, the width distribution of the relaxation processes involved in the
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